| L(s) = 1 | − 3-s + 3·5-s + 5·7-s − 2·9-s − 4·11-s − 5·13-s − 3·15-s − 11·17-s − 3·19-s − 5·21-s + 9·23-s + 6·25-s − 3·27-s − 3·29-s + 14·31-s + 4·33-s + 15·35-s − 14·37-s + 5·39-s − 10·41-s − 10·43-s − 6·45-s + 4·49-s + 11·51-s + 7·53-s − 12·55-s + 3·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s + 1.88·7-s − 2/3·9-s − 1.20·11-s − 1.38·13-s − 0.774·15-s − 2.66·17-s − 0.688·19-s − 1.09·21-s + 1.87·23-s + 6/5·25-s − 0.577·27-s − 0.557·29-s + 2.51·31-s + 0.696·33-s + 2.53·35-s − 2.30·37-s + 0.800·39-s − 1.56·41-s − 1.52·43-s − 0.894·45-s + 4/7·49-s + 1.54·51-s + 0.961·53-s − 1.61·55-s + 0.397·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) |
| 19 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + T + p T^{2} + 8 T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 5 T + 3 p T^{2} - 62 T^{3} + 3 p^{2} T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 4 T + 13 T^{2} + 24 T^{3} + 13 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 5 T + 17 T^{2} + 24 T^{3} + 17 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 11 T + 83 T^{2} + 394 T^{3} + 83 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 9 T + 53 T^{2} - 254 T^{3} + 53 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 3 T + 15 T^{2} + 66 T^{3} + 15 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 14 T + 125 T^{2} - 804 T^{3} + 125 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 14 T + 157 T^{2} + 1016 T^{3} + 157 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 10 T + 79 T^{2} + 348 T^{3} + 79 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 10 T + 41 T^{2} + 12 T^{3} + 41 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 101 T^{2} - 64 T^{3} + 101 p T^{4} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 7 T + 145 T^{2} - 744 T^{3} + 145 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 7 T + 185 T^{2} + 818 T^{3} + 185 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 20 T + 215 T^{2} + 1800 T^{3} + 215 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + T + p T^{2} + 664 T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 8 T + 133 T^{2} - 624 T^{3} + 133 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 13 T + 155 T^{2} + 1398 T^{3} + 155 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 14 T + 181 T^{2} + 2228 T^{3} + 181 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 26 T + 441 T^{2} + 4668 T^{3} + 441 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 18 T + 335 T^{2} - 3244 T^{3} + 335 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 6 T + 17 T^{2} - 1144 T^{3} + 17 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.33381280502520934848432098188, −7.30080376077402126920512200752, −6.98825402927041895479219513650, −6.77683606269955075697780742848, −6.43657434991135070680741895311, −6.33918487124606852072154851258, −6.13770433104012937758193151092, −5.72289922222466275001642718572, −5.48262033806665469354359955986, −5.24433629181362530062002852270, −4.94539963299939366032658138719, −4.89426276272825863569675698598, −4.79418331645031123147628770715, −4.60620831537287531706031345170, −4.21056578965318731338212295740, −3.90814921386339962904361764522, −3.33432258732142566730945570160, −3.09862536582831536032402750829, −2.78272999301739589750607082250, −2.49277613862242872299506300480, −2.38698190335972048917653610342, −1.92480070103248348434855175060, −1.78608210830072041461264581885, −1.35990036725142598241340569879, −1.22028073388181966980160800317, 0, 0, 0,
1.22028073388181966980160800317, 1.35990036725142598241340569879, 1.78608210830072041461264581885, 1.92480070103248348434855175060, 2.38698190335972048917653610342, 2.49277613862242872299506300480, 2.78272999301739589750607082250, 3.09862536582831536032402750829, 3.33432258732142566730945570160, 3.90814921386339962904361764522, 4.21056578965318731338212295740, 4.60620831537287531706031345170, 4.79418331645031123147628770715, 4.89426276272825863569675698598, 4.94539963299939366032658138719, 5.24433629181362530062002852270, 5.48262033806665469354359955986, 5.72289922222466275001642718572, 6.13770433104012937758193151092, 6.33918487124606852072154851258, 6.43657434991135070680741895311, 6.77683606269955075697780742848, 6.98825402927041895479219513650, 7.30080376077402126920512200752, 7.33381280502520934848432098188
